Sketch conic r = 12/(3+2cosθ)

Divide the numerator and denominator by 3 to match r = l/(1 + e cosθ):

r = 12/(3 + 2cosθ) = 4 / (1 + (2/3)cosθ)

So e = 2/3 < 1: the conic is an ellipse, with the pole at one focus and semi-latus rectum l = 4.

Key facts

• ed = 4, so d = 6: directrix x = 6.
• θ = 0: r = 12/5 = 2.4, vertex (2.4, 0).
• θ = π: r = 12, vertex (−12, 0).
• θ = π/2: r = 4, so the latus rectum ends are (0, ±4).
• Major axis 2a = 2.4 + 12 = 14.4, so a = 36/5 = 7.2.
• Centre = (−24/5, 0) = (−4.8, 0); ae = 4.8, so the other focus is (−9.6, 0).
• b² = a²(1 − e²) = (1296/25)(5/9) = 144/5, so b = 12/√5 ≈ 5.37.

                              y
                              |          x = 6
              .  -  -  -  -  -|-  .        |
          .'                  | (0,4) '.   |
---------(-----F2-------C-----F1-------)---|------ x
       -12   -9.6     -4.8    0       2.4  6
          '.                  | (0,-4).'   |
              '  -  -  -  -  -|-  '        |
                              |

The curve is an ellipse stretched along the x-axis from x = −12 to x = 2.4, with centre C(−4.8, 0), foci F1 at the origin (the pole) and F2(−9.6, 0), passing through (0, ±4), and with the directrix x = 6 on the right.